Category: mathematics

Kathleen Howell is an American scientist and aerospace engineer. Her contributions to the theory of dynamical systems have been applied to spacecraft trajectory design, which led to the use of near-rectilinear halo orbit (NRHO) in various NASA space missions.

Unlike an ordinary flat orbit, an NRHO can be slightly warped. Further, it stands on end, almost perpendicular to an ordinary orbit – hence “near rectilinear”.

NASA have decided that an NRHO would be an ideal place to put the Lunar Orbital Platform-Gateway, which is a planned way station for future human flights to the Moon and eventually Mars. The plan is for the Gateway’s circuit to pass tight over the Moon’s north pole at high speed and more slowly below the south pole, because of the greater distance from the moon.

“Imagine moving your hand in circles, as if washing a window, while you walk forward. Except you’re making hand circles around the moon while walking around Earth.” – Bloomberg

Although this orbit seems to be an ordinary circuit of the moon, it’s actually part of a family of orbits, centred on an empty point, called L2 (or Lagrange Point 2). Here, around 45,000 miles beyond the far side of the Moon, the gravitational forces of the Earth and the moon are in balance with the centrifugal forces on the spacecraft.

Although we are taught in school that orbits must be around something, it is quite possible to orbit around nothing, so long as that ‘nothing’ is a Lagrange point.

“It is elegant and very rich. All the forces come together to produce an unexpected path through space” – Howell

Howell’s work build on an 18th century discovery, by Euler, who theorised that for any pair of orbiting bodies, there are 3 points in space where gravitational and centrifugal forces balance precisely. In 1772, Lagrange found two more such spots. All five are now known as Lagrange points.

Source: Wikipedia

In 2017 Kathleen Howell was elected to National Academy of Engineering “for contributions in dynamical systems theory and invariant manifolds culminating in optimal interplanetary trajectories and the Interplanetary Superhighway“.

The four winners of the Fields Medal for 2018 have been announced. The Fields Medal is awarded every 4 years at an international gathering of mathematicians and is considered the Nobel Prize for Mathematics. However, there is one key difference: recipients must be 40 years old or younger.

This years recipients were announced on Wednesday 1st August at the International Congress of Mathematicians in Rio de Janeiro. They are:

Caucher Birker, 40, of the University of Cambridge in England: “for his proof of the boundedness of Fano varieties and for contributions to the minimal model program.”

Allesio Figalli, 34, of the Swiss Federal Institute of Technology in Zurich: “for his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability.”

Akshay Venkatesh, 36, of the Institute for Advanced Study in Princeton and Stanford University in California: “for his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects.” Click here for more information.

Peter Scholze, 30 (one of the youngest recipients), of the University of Bonn in Germany: “for transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to galois representations and for the development of new cohomology theories.” Click here for more information.

Probability and Statistics

Gauss introduced what is now known as the Gaussian distribution: he showed how probability can be represented by a bell-shaped curve, with peaks around the mean when falls off quickly towards plus or minus infinity.

Source: Wikipedia

He also created the Gaussian function: a function of the form

for arbitrary real constants a, b and c.

Modular Arithmetic

The modern approach to modular arithmetic was developed by Gauss in his book Disquisitiones Arithmeticae, published in 1801. This now has application in number theory, abstract algebra, computer science, cryptography, and even in visual and musical art.

Surveying

Whilst doing a surveying job for the Royal House of Hanover in the years after 1818, Gauss was also looking into the shape of the Earth and started to question what the shape of space itself was. This led him to question Euclidean geometry – one of the central tenets of the whole mathematics, which premised a flat universe, rather than a curved one. He later claimed that as early as 1800 he had already started to consider types of non-Euclidean geometry (where the parallel axiom does not hold), which were consistent and free of contradiction. However, to avoid controversy, he did not publish anything in this area and left the field open to Bolyai and Lobachevsky, although he is still considered by some to be the pioneer of non-Euclidean geometry.

This survey work also fuelled Gauss’ interest in differential geometry, which uses differential calculus to study problems in geometry involving curves and surfaces. He developed what has become known as Gaussian curvature. This is an intrinsic measure of curvature that depends only on how distances are measured on the surface, not on the way it is embedded in space.

Source: shellbuckling.com

His achievements during these years, however, was not only limited to pure mathematics. He invented the heliotrope, which is an instrument that uses a mirror to reflect sunlight over great distances to mark positions in a land survey.

Heliotrope | Source: Wikipedia

All in all, this period of time was one of the most fruitful periods of his academic life; he published over 70 papers between 1820 and 1830.

In later years, he worked with Wilhelm Weber to make measurements of the Earth’s magnetic field, and invented the first electric telegraph.

Although Gauss made contributions in many fields of mathematics, number theory was his favourite. He said that

“mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics.”

A way in which Gauss revolutionised number theory was his work with complex numbers.

Gauss gave the first clear exposition of complex numbers and of the investigation of functions of complex variables. Although imaginary numbers had been used since the 16th century to solve equations that couldn’t be solved any other way, and although Euler made huge progress in this field in the 18th century, there was still no clear idea as to how imaginary numbers were connected with real numbers until early 19th century. Gauss was not the first to picture complex numbers graphically (Robert Argand produced the Argand diagram in 1806). However, Gauss was the one who popularised this idea and introduced the standard notation a + bi. Hence, the study of complex numbers received a great expansion allowing its full potential to be unleashed.

Furthermore, at the age of 22 he proved the Fundamental Theorem of Algebra which states:

Every non-constant single-variable polynomial over the complex numbers has at least one root.

This shows that the field of complex numbers is algebraically closed, unlike the real numbers.

Gauss also had a strong interest in astronomy, and was the Director of the astronomical observatory in Göttingen. When Ceres was in the process of being identifies in the late 17th century, Gauss made a prediction of its position. This prediction was very different from those of other astronomers, but when Ceres was discovered in 1801, it was almost exactly where Gauss had predicted. This was one of the first applications of the least squares approximation method, and Gauss claimed to have done the logarithmic calculations in his head.

I decided to start a new series on influential mathematicians, starting with Gauss, one of my personal favourites. Carl Friedrich Gauss (1777-1855) is considered to be one of the greatest mathematicians in the 19th century, and is sometimes referred to as the “Prince of Mathematics”.

His discoveries influenced and left a lasting mark in a variety of different areas, including number theory, astronomy, geodesy, and physics, particularly the study of electromagnetism.

Born in Brunswick, Germany to poor, working class parents, he was discouraged from attending school from his father, a gardner and brick-layer, who expected Gauss would follow one of the family trades. However, Gauss’ mother and uncle recognised Gauss’ early genius and knew he must develop this gift with a proper education.

In arithmetic class, at the age of 10, Gauss showed his skills as a maths prodigy. A well known anecdote about Gauss and his early school education is about when the strict schoolmaster gave the following assignment:

“Write down all the whole numbers from 1 to 100 and add up their sum.”

They expected this assignment to take a while to complete but after a few seconds, to the teacher’s surprise, Carl placed his slate on the desk in front of the teacher, showing he was done with the question. His other classmates took a much longer time to complete the assignment. At the end of class time, although most other students answers were wrong, Gauss’ was correct: 5050. Carl then explained to the teacher that he found the result as he could see that 1+100 = 101, 2+99=101, etc. So he could find 50 pairs of numbers that each add up to 101, and so 50*101 = 5050. I don’t know about you but I definitely could not come up with this sort of argument at the age of 10…

Although his family was poor, Gauss’ intellectual abilities drew the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum at the age of 15, and then to the University of Göttingen – a very prestigious university – where he stayed from 1795 to 1798. During this period, Gauss discovered many important theorems.

Prime Numbers

No pattern had previously been found in the occurrence of prime numbers until Gauss. Although the occurrence of the primes seems to be completely random, by approaching the problem from a different angle and graphing the incidence of primes as the numbers increased, he noticed a rough trend: as numbers increased by 10, the probability of the numbers reduced by a factor of around 2. However, as his method only gave him an approximation, and as he could not definitively prove his findings, he kept them a secret until much later in his life.

1796

1796 is known as Gauss’ “annus mirabilis” (means “wonderful year” and is used to refer to several years during which events of major importance are remembered). In 1796:

Gauss constructed a regular 17-sided heptadecagon, which had previously been unknown, using only a ruler and a compass. This was a major advance in geometry since the time of the Greeks.

Gauss formulated this prime number theorem on the distribution of prime numbers among the integers, which states that Here is the number of primes less than or equal to n. We can also write .

Gauss proved that every positive integer can be represented as the sum of at most 3 triangular number

The Brouwer Fixed Point Theorem is a result in topology that has proven to be extremely useful in mathematics.

2D

Suppose you take two sheets of paper with one lying directly above the other. If you crumple the top sheet and place it on top of the other sheet, then Brouwer’s theorem states that there is at least one point that has not moved, i.e. there is at least one point on the top sheet that is directly above the corresponding point on the bottom sheet.

3D

Take a cup of coffee, and mix it around. The theorem states that after mixing there must be some point in the coffee, which is in the exact spot that it was before the mixing. Furthermore, if you try to move that point out of its original position then you will inevitably move another point into its original position.

Formally

A continuous function from an n-ball into an n-ball must have a fixed point. Note that continuity of the function is essential.